Is a Beginningless Past Actually Infinite?

Dear Dr Craig,

In the last talk of the conference Formal Methods in the Epistemology of Religion (June 2009, Belgium), the famous European atheist Herman Philipse raised (in the introduction) a minor worry which I've shared with him for a long time: your Kalam arguments have problematic premise that beginningless causal chain would be (or form) an actually infinite set.

Indeed, if you are a presentist, why should you say that if the past was actually infinite, then the past intervals would form an actually infinite set (as is assumed in Kalam cosmological arguments)?

If only the present temporal realm is real (as the presentist claims), then there are no past intervals to form a set.

If you adopted degree presentism -- saying that the present is more real than the past and the future which are nevertheless also real, but to a lesser degree -- then you could assume multiple past intervals. Yet, then you should assume multiple future intervals, too -- and so you could run some Kalam argument against the infinity of the future. But it seems you want to avoid the conclusion that the future is finite. At least, such a conclusion would generate a problem for eternal heaven and hell: time is essential for humans, and it also does not make sense to say, "after this temporal life, there will be a timeless existence."

Any thoughts on the way out?

Thank you.

Vlastimil

Good to hear from you, Vlastimil! Say “Hi” to Paula! It’s gratifying, isn’t it, to see that these issues are finally appearing on the European horizon? I’ve just been invited to contribute essays on the kalam cosmological argument to two German books of essays on the arguments of natural theology. To have such issues even discussed, even if critically, as in the conference you recount, is progress.

Strictly speaking, I wouldn’t say, as you put it, that a “beginningless causal chain would be (or form) an actually infinite set.” Sets, if they exist, are abstract objects and so should not be identified with the series of events in time. Using what I would regard as the useful fiction of a set, I suppose we could say that the set of past events is an infinite set if the series of past events is beginningless. But I prefer simply to say that if the temporal series of events is beginningless, then the number of past events is infinite or that there has occurred an infinite number of past events.

So saying seems to me unproblematic. The problems that arise in this connection are, I think, due to the difficulty of capturing various intuitive notions linguistically because of the presence of tense. For example, there seems to be an intuitive difference between the presentist, who thinks that only the present moment of time exists, and so-called eternalists, who think that all moments of time exist. But many philosophers have noted how difficult it is to express this difference linguistically. For if “exists” is present-tensed, than eternalists agree that the only moment of time that exists is the present! For that is just to say that only the present moment presently exists. On the other hand, if “exists” is tenseless, then the presentist agrees that all moments of time exist! For that just is to say that every moment of time exists at some time or other. Philosophers have resorted to linguistic contortionism in order to express the difference between presentism and eternalism. I think that it’s obvious that these are two different ontologies, even if our linguistic resources fail us in our attempts to express the difference in sentences.

On an eternalist ontology, since all events are equally real, there can be no question that a beginningless temporal regress of events is composed of an actually infinite number of events. Since all events are equally real, the fact that they exist (tenselessly) at different times loses any significance. The question, then, is whether events’ temporal distribution over the past on a presentist ontology precludes our saying that the number of events in a beginningless series of events is actually infinite.

Now we may take it as a datum that the presentist can accurately count things that have existed but no longer exist. He knows, for example, how many U.S. presidents there have been up through the present incumbent, what day of the month it is, how many weeks it has been since his last haircut, and so forth. He knows how old his children are and can reckon how many billion years have elapsed since the Big Bang. The non-existence of such things or events is no hindrance to their being enumerated. Indeed, any obstacle here is merely epistemic, for aside from considerations of vagueness there must be a certain number of such things. So in a beginningless series of past events of equal duration, the number of past events must be infinite, for it is larger than any natural number. But then the number of past events must be ℵ0 (the first transfinite cardinal number), for ∞ (signifying a potential infinite) is not a number but an ideal limit.

The question then arises whether on presentism the series of future events, if time is endless, is not also actually infinite. Intuitively, it seems clear that the situation is not symmetrical, but this is notoriously difficult to express linguistically. It might rightly be pointed out that on presentism there are no future events and so no series of future events. Therefore, the number of future events is simply zero, not ℵ0. (By this statement one means, not that there are future events, and their number is 0, but that there just are no future events.) But on presentism, the past is as unreal as the future and therefore the number of past events could with equal justification be said to be zero.

It might be said that at least there have been past events, and so they can be numbered. But by the same token there will be future events, so why can they not be numbered? Accordingly, one might be tempted to say that in an endless future there will be an actually infinite number of events, just as in a beginningless past there have been an actually infinite number of events. But in a sense that assertion is false; for there never will be an actually infinite number of events, since it is impossible to count to infinity. The only sense in which there will be an infinite number of events is that the series of events will go toward infinity as a limit.

But that is the concept of a potential infinite, not an actual infinite. Here the objectivity of temporal becoming makes itself felt. For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as potentially infinite, that is to say, finite but indefinitely increasing toward infinity as a limit. The situation, significantly, is not symmetrical: as we have seen, the series of events earlier than any arbitrarily selected future event cannot properly be regarded as potentially infinite. So when we say that the number of past events is infinite, we mean that prior to today ℵ0 events have elapsed. But when we say that the number of future events is infinite, we do not mean that ℵ0 events will elapse, for that is false.

Ironically, then, it turns out that the series of future events cannot be actually infinite regardless of the infinity of the past or the possibility of an actual infinite, for it is the objectivity of temporal becoming that makes the future potentially infinite only.